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Properties

Label 280.3.c.a.69.1
Level $280$
Weight $3$
Character 280.69
Self dual yes
Analytic conductor $7.629$
Analytic rank $0$
Dimension $1$
CM discriminant -280
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,3,Mod(69,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.69");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 280.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.62944740209\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 69.1
Character \(\chi\) \(=\) 280.69

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +14.0000 q^{14} +16.0000 q^{16} +6.00000 q^{17} -18.0000 q^{18} +18.0000 q^{19} -20.0000 q^{20} +25.0000 q^{25} -28.0000 q^{28} -32.0000 q^{32} -12.0000 q^{34} +35.0000 q^{35} +36.0000 q^{36} +66.0000 q^{37} -36.0000 q^{38} +40.0000 q^{40} +54.0000 q^{43} -45.0000 q^{45} +66.0000 q^{47} +49.0000 q^{49} -50.0000 q^{50} +34.0000 q^{53} +56.0000 q^{56} -62.0000 q^{59} +102.000 q^{61} -63.0000 q^{63} +64.0000 q^{64} +6.00000 q^{67} +24.0000 q^{68} -70.0000 q^{70} -138.000 q^{71} -72.0000 q^{72} -106.000 q^{73} -132.000 q^{74} +72.0000 q^{76} -122.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -30.0000 q^{85} -108.000 q^{86} +90.0000 q^{90} -132.000 q^{94} -90.0000 q^{95} +166.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 4.00000 1.00000
\(5\) −5.00000 −1.00000
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) −8.00000 −1.00000
\(9\) 9.00000 1.00000
\(10\) 10.0000 1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 14.0000 1.00000
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 6.00000 0.352941 0.176471 − 0.984306i \(-0.443532\pi\)
0.176471 + 0.984306i \(0.443532\pi\)
\(18\) −18.0000 −1.00000
\(19\) 18.0000 0.947368 0.473684 − 0.880695i \(-0.342924\pi\)
0.473684 + 0.880695i \(0.342924\pi\)
\(20\) −20.0000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −28.0000 −1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −32.0000 −1.00000
\(33\) 0 0
\(34\) −12.0000 −0.352941
\(35\) 35.0000 1.00000
\(36\) 36.0000 1.00000
\(37\) 66.0000 1.78378 0.891892 − 0.452249i \(-0.149378\pi\)
0.891892 + 0.452249i \(0.149378\pi\)
\(38\) −36.0000 −0.947368
\(39\) 0 0
\(40\) 40.0000 1.00000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 54.0000 1.25581 0.627907 − 0.778288i \(-0.283912\pi\)
0.627907 + 0.778288i \(0.283912\pi\)
\(44\) 0 0
\(45\) −45.0000 −1.00000
\(46\) 0 0
\(47\) 66.0000 1.40426 0.702128 − 0.712051i \(-0.252234\pi\)
0.702128 + 0.712051i \(0.252234\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) −50.0000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 34.0000 0.641509 0.320755 − 0.947162i \(-0.396063\pi\)
0.320755 + 0.947162i \(0.396063\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 56.0000 1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) −62.0000 −1.05085 −0.525424 − 0.850841i \(-0.676093\pi\)
−0.525424 + 0.850841i \(0.676093\pi\)
\(60\) 0 0
\(61\) 102.000 1.67213 0.836066 − 0.548630i \(-0.184850\pi\)
0.836066 + 0.548630i \(0.184850\pi\)
\(62\) 0 0
\(63\) −63.0000 −1.00000
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 0.0895522 0.0447761 − 0.998997i \(-0.485743\pi\)
0.0447761 + 0.998997i \(0.485743\pi\)
\(68\) 24.0000 0.352941
\(69\) 0 0
\(70\) −70.0000 −1.00000
\(71\) −138.000 −1.94366 −0.971831 − 0.235679i \(-0.924269\pi\)
−0.971831 + 0.235679i \(0.924269\pi\)
\(72\) −72.0000 −1.00000
\(73\) −106.000 −1.45205 −0.726027 − 0.687666i \(-0.758635\pi\)
−0.726027 + 0.687666i \(0.758635\pi\)
\(74\) −132.000 −1.78378
\(75\) 0 0
\(76\) 72.0000 0.947368
\(77\) 0 0
\(78\) 0 0
\(79\) −122.000 −1.54430 −0.772152 − 0.635438i \(-0.780820\pi\)
−0.772152 + 0.635438i \(0.780820\pi\)
\(80\) −80.0000 −1.00000
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −30.0000 −0.352941
\(86\) −108.000 −1.25581
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 90.0000 1.00000
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −132.000 −1.40426
\(95\) −90.0000 −0.947368
\(96\) 0 0
\(97\) 166.000 1.71134 0.855670 − 0.517522i \(-0.173145\pi\)
0.855670 + 0.517522i \(0.173145\pi\)
\(98\) −98.0000 −1.00000
\(99\) 0 0
\(100\) 100.000 1.00000
\(101\) 22.0000 0.217822 0.108911 − 0.994052i \(-0.465264\pi\)
0.108911 + 0.994052i \(0.465264\pi\)
\(102\) 0 0
\(103\) −46.0000 −0.446602 −0.223301 − 0.974750i \(-0.571683\pi\)
−0.223301 + 0.974750i \(0.571683\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −68.0000 −0.641509
\(107\) −74.0000 −0.691589 −0.345794 − 0.938310i \(-0.612391\pi\)
−0.345794 + 0.938310i \(0.612391\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −112.000 −1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 124.000 1.05085
\(119\) −42.0000 −0.352941
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) −204.000 −1.67213
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −1.00000
\(126\) 126.000 1.00000
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −128.000 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 242.000 1.84733 0.923664 − 0.383203i \(-0.125179\pi\)
0.923664 + 0.383203i \(0.125179\pi\)
\(132\) 0 0
\(133\) −126.000 −0.947368
\(134\) −12.0000 −0.0895522
\(135\) 0 0
\(136\) −48.0000 −0.352941
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −222.000 −1.59712 −0.798561 − 0.601914i \(-0.794405\pi\)
−0.798561 + 0.601914i \(0.794405\pi\)
\(140\) 140.000 1.00000
\(141\) 0 0
\(142\) 276.000 1.94366
\(143\) 0 0
\(144\) 144.000 1.00000
\(145\) 0 0
\(146\) 212.000 1.45205
\(147\) 0 0
\(148\) 264.000 1.78378
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 22.0000 0.145695 0.0728477 − 0.997343i \(-0.476791\pi\)
0.0728477 + 0.997343i \(0.476791\pi\)
\(152\) −144.000 −0.947368
\(153\) 54.0000 0.352941
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 244.000 1.54430
\(159\) 0 0
\(160\) 160.000 1.00000
\(161\) 0 0
\(162\) −162.000 −1.00000
\(163\) −186.000 −1.14110 −0.570552 − 0.821261i \(-0.693271\pi\)
−0.570552 + 0.821261i \(0.693271\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 306.000 1.83234 0.916168 − 0.400795i \(-0.131266\pi\)
0.916168 + 0.400795i \(0.131266\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 60.0000 0.352941
\(171\) 162.000 0.947368
\(172\) 216.000 1.25581
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −175.000 −1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −180.000 −1.00000
\(181\) −138.000 −0.762431 −0.381215 − 0.924486i \(-0.624494\pi\)
−0.381215 + 0.924486i \(0.624494\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −330.000 −1.78378
\(186\) 0 0
\(187\) 0 0
\(188\) 264.000 1.40426
\(189\) 0 0
\(190\) 180.000 0.947368
\(191\) 102.000 0.534031 0.267016 − 0.963692i \(-0.413962\pi\)
0.267016 + 0.963692i \(0.413962\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) −332.000 −1.71134
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) −254.000 −1.28934 −0.644670 − 0.764461i \(-0.723005\pi\)
−0.644670 + 0.764461i \(0.723005\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −200.000 −1.00000
\(201\) 0 0
\(202\) −44.0000 −0.217822
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 92.0000 0.446602
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 136.000 0.641509
\(213\) 0 0
\(214\) 148.000 0.691589
\(215\) −270.000 −1.25581
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 194.000 0.869955 0.434978 − 0.900441i \(-0.356756\pi\)
0.434978 + 0.900441i \(0.356756\pi\)
\(224\) 224.000 1.00000
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 438.000 1.91266 0.956332 − 0.292283i \(-0.0944149\pi\)
0.956332 + 0.292283i \(0.0944149\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −330.000 −1.40426
\(236\) −248.000 −1.05085
\(237\) 0 0
\(238\) 84.0000 0.352941
\(239\) 198.000 0.828452 0.414226 − 0.910174i \(-0.364052\pi\)
0.414226 + 0.910174i \(0.364052\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −242.000 −1.00000
\(243\) 0 0
\(244\) 408.000 1.67213
\(245\) −245.000 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 250.000 1.00000
\(251\) 2.00000 0.00796813 0.00398406 − 0.999992i \(-0.498732\pi\)
0.00398406 + 0.999992i \(0.498732\pi\)
\(252\) −252.000 −1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 486.000 1.89105 0.945525 − 0.325549i \(-0.105549\pi\)
0.945525 + 0.325549i \(0.105549\pi\)
\(258\) 0 0
\(259\) −462.000 −1.78378
\(260\) 0 0
\(261\) 0 0
\(262\) −484.000 −1.84733
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −170.000 −0.641509
\(266\) 252.000 0.947368
\(267\) 0 0
\(268\) 24.0000 0.0895522
\(269\) 358.000 1.33086 0.665428 − 0.746462i \(-0.268249\pi\)
0.665428 + 0.746462i \(0.268249\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 96.0000 0.352941
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −414.000 −1.49458 −0.747292 − 0.664495i \(-0.768647\pi\)
−0.747292 + 0.664495i \(0.768647\pi\)
\(278\) 444.000 1.59712
\(279\) 0 0
\(280\) −280.000 −1.00000
\(281\) −558.000 −1.98577 −0.992883 − 0.119098i \(-0.962000\pi\)
−0.992883 + 0.119098i \(0.962000\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −552.000 −1.94366
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −288.000 −1.00000
\(289\) −253.000 −0.875433
\(290\) 0 0
\(291\) 0 0
\(292\) −424.000 −1.45205
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 310.000 1.05085
\(296\) −528.000 −1.78378
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −378.000 −1.25581
\(302\) −44.0000 −0.145695
\(303\) 0 0
\(304\) 288.000 0.947368
\(305\) −510.000 −1.67213
\(306\) −108.000 −0.352941
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 374.000 1.19489 0.597444 − 0.801911i \(-0.296183\pi\)
0.597444 + 0.801911i \(0.296183\pi\)
\(314\) 0 0
\(315\) 315.000 1.00000
\(316\) −488.000 −1.54430
\(317\) 626.000 1.97476 0.987382 − 0.158358i \(-0.0506201\pi\)
0.987382 + 0.158358i \(0.0506201\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −320.000 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 108.000 0.334365
\(324\) 324.000 1.00000
\(325\) 0 0
\(326\) 372.000 1.14110
\(327\) 0 0
\(328\) 0 0
\(329\) −462.000 −1.40426
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 594.000 1.78378
\(334\) −612.000 −1.83234
\(335\) −30.0000 −0.0895522
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −338.000 −1.00000
\(339\) 0 0
\(340\) −120.000 −0.352941
\(341\) 0 0
\(342\) −324.000 −0.947368
\(343\) −343.000 −1.00000
\(344\) −432.000 −1.25581
\(345\) 0 0
\(346\) 0 0
\(347\) 566.000 1.63112 0.815562 − 0.578670i \(-0.196428\pi\)
0.815562 + 0.578670i \(0.196428\pi\)
\(348\) 0 0
\(349\) 198.000 0.567335 0.283668 − 0.958923i \(-0.408449\pi\)
0.283668 + 0.958923i \(0.408449\pi\)
\(350\) 350.000 1.00000
\(351\) 0 0
\(352\) 0 0
\(353\) −666.000 −1.88669 −0.943343 − 0.331820i \(-0.892337\pi\)
−0.943343 + 0.331820i \(0.892337\pi\)
\(354\) 0 0
\(355\) 690.000 1.94366
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 438.000 1.22006 0.610028 − 0.792380i \(-0.291158\pi\)
0.610028 + 0.792380i \(0.291158\pi\)
\(360\) 360.000 1.00000
\(361\) −37.0000 −0.102493
\(362\) 276.000 0.762431
\(363\) 0 0
\(364\) 0 0
\(365\) 530.000 1.45205
\(366\) 0 0
\(367\) 706.000 1.92371 0.961853 − 0.273567i \(-0.0882036\pi\)
0.961853 + 0.273567i \(0.0882036\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 660.000 1.78378
\(371\) −238.000 −0.641509
\(372\) 0 0
\(373\) −606.000 −1.62466 −0.812332 − 0.583195i \(-0.801802\pi\)
−0.812332 + 0.583195i \(0.801802\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −528.000 −1.40426
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −360.000 −0.947368
\(381\) 0 0
\(382\) −204.000 −0.534031
\(383\) −606.000 −1.58225 −0.791123 − 0.611657i \(-0.790503\pi\)
−0.791123 + 0.611657i \(0.790503\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 486.000 1.25581
\(388\) 664.000 1.71134
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −392.000 −1.00000
\(393\) 0 0
\(394\) 508.000 1.28934
\(395\) 610.000 1.54430
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) −318.000 −0.793017 −0.396509 − 0.918031i \(-0.629778\pi\)
−0.396509 + 0.918031i \(0.629778\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 88.0000 0.217822
\(405\) −405.000 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −184.000 −0.446602
\(413\) 434.000 1.05085
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −782.000 −1.86635 −0.933174 − 0.359424i \(-0.882973\pi\)
−0.933174 + 0.359424i \(0.882973\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 594.000 1.40426
\(424\) −272.000 −0.641509
\(425\) 150.000 0.352941
\(426\) 0 0
\(427\) −714.000 −1.67213
\(428\) −296.000 −0.691589
\(429\) 0 0
\(430\) 540.000 1.25581
\(431\) 582.000 1.35035 0.675174 − 0.737658i \(-0.264069\pi\)
0.675174 + 0.737658i \(0.264069\pi\)
\(432\) 0 0
\(433\) −506.000 −1.16859 −0.584296 − 0.811541i \(-0.698629\pi\)
−0.584296 + 0.811541i \(0.698629\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 374.000 0.844244 0.422122 − 0.906539i \(-0.361285\pi\)
0.422122 + 0.906539i \(0.361285\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −388.000 −0.869955
\(447\) 0 0
\(448\) −448.000 −1.00000
\(449\) −222.000 −0.494432 −0.247216 − 0.968960i \(-0.579516\pi\)
−0.247216 + 0.968960i \(0.579516\pi\)
\(450\) −450.000 −1.00000
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −876.000 −1.91266
\(459\) 0 0
\(460\) 0 0
\(461\) −698.000 −1.51410 −0.757050 − 0.653357i \(-0.773360\pi\)
−0.757050 + 0.653357i \(0.773360\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −42.0000 −0.0895522
\(470\) 660.000 1.40426
\(471\) 0 0
\(472\) 496.000 1.05085
\(473\) 0 0
\(474\) 0 0
\(475\) 450.000 0.947368
\(476\) −168.000 −0.352941
\(477\) 306.000 0.641509
\(478\) −396.000 −0.828452
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) −830.000 −1.71134
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −816.000 −1.67213
\(489\) 0 0
\(490\) 490.000 1.00000
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 966.000 1.94366
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −500.000 −1.00000
\(501\) 0 0
\(502\) −4.00000 −0.00796813
\(503\) −366.000 −0.727634 −0.363817 − 0.931470i \(-0.618527\pi\)
−0.363817 + 0.931470i \(0.618527\pi\)
\(504\) 504.000 1.00000
\(505\) −110.000 −0.217822
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 998.000 1.96071 0.980354 − 0.197248i \(-0.0632004\pi\)
0.980354 + 0.197248i \(0.0632004\pi\)
\(510\) 0 0
\(511\) 742.000 1.45205
\(512\) −512.000 −1.00000
\(513\) 0 0
\(514\) −972.000 −1.89105
\(515\) 230.000 0.446602
\(516\) 0 0
\(517\) 0 0
\(518\) 924.000 1.78378
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 968.000 1.84733
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 340.000 0.641509
\(531\) −558.000 −1.05085
\(532\) −504.000 −0.947368
\(533\) 0 0
\(534\) 0 0
\(535\) 370.000 0.691589
\(536\) −48.0000 −0.0895522
\(537\) 0 0
\(538\) −716.000 −1.33086
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −192.000 −0.352941
\(545\) 0 0
\(546\) 0 0
\(547\) −954.000 −1.74406 −0.872029 − 0.489454i \(-0.837196\pi\)
−0.872029 + 0.489454i \(0.837196\pi\)
\(548\) 0 0
\(549\) 918.000 1.67213
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 854.000 1.54430
\(554\) 828.000 1.49458
\(555\) 0 0
\(556\) −888.000 −1.59712
\(557\) 146.000 0.262118 0.131059 − 0.991375i \(-0.458162\pi\)
0.131059 + 0.991375i \(0.458162\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 560.000 1.00000
\(561\) 0 0
\(562\) 1116.00 1.98577
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −567.000 −1.00000
\(568\) 1104.00 1.94366
\(569\) 18.0000 0.0316344 0.0158172 − 0.999875i \(-0.494965\pi\)
0.0158172 + 0.999875i \(0.494965\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 576.000 1.00000
\(577\) −1114.00 −1.93068 −0.965338 − 0.261003i \(-0.915947\pi\)
−0.965338 + 0.261003i \(0.915947\pi\)
\(578\) 506.000 0.875433
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 848.000 1.45205
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −620.000 −1.05085
\(591\) 0 0
\(592\) 1056.00 1.78378
\(593\) −186.000 −0.313659 −0.156830 − 0.987626i \(-0.550127\pi\)
−0.156830 + 0.987626i \(0.550127\pi\)
\(594\) 0 0
\(595\) 210.000 0.352941
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 918.000 1.53255 0.766277 − 0.642510i \(-0.222107\pi\)
0.766277 + 0.642510i \(0.222107\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 756.000 1.25581
\(603\) 54.0000 0.0895522
\(604\) 88.0000 0.145695
\(605\) −605.000 −1.00000
\(606\) 0 0
\(607\) −1054.00 −1.73641 −0.868204 − 0.496207i \(-0.834726\pi\)
−0.868204 + 0.496207i \(0.834726\pi\)
\(608\) −576.000 −0.947368
\(609\) 0 0
\(610\) 1020.00 1.67213
\(611\) 0 0
\(612\) 216.000 0.352941
\(613\) −1086.00 −1.77162 −0.885808 − 0.464053i \(-0.846395\pi\)
−0.885808 + 0.464053i \(0.846395\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1182.00 −1.90953 −0.954766 − 0.297359i \(-0.903894\pi\)
−0.954766 + 0.297359i \(0.903894\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) −748.000 −1.19489
\(627\) 0 0
\(628\) 0 0
\(629\) 396.000 0.629571
\(630\) −630.000 −1.00000
\(631\) −1258.00 −1.99366 −0.996830 − 0.0795556i \(-0.974650\pi\)
−0.996830 + 0.0795556i \(0.974650\pi\)
\(632\) 976.000 1.54430
\(633\) 0 0
\(634\) −1252.00 −1.97476
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1242.00 −1.94366
\(640\) 640.000 1.00000
\(641\) 162.000 0.252730 0.126365 − 0.991984i \(-0.459669\pi\)
0.126365 + 0.991984i \(0.459669\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −216.000 −0.334365
\(647\) 1266.00 1.95672 0.978362 − 0.206902i \(-0.0663381\pi\)
0.978362 + 0.206902i \(0.0663381\pi\)
\(648\) −648.000 −1.00000
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −744.000 −1.14110
\(653\) −46.0000 −0.0704441 −0.0352221 − 0.999380i \(-0.511214\pi\)
−0.0352221 + 0.999380i \(0.511214\pi\)
\(654\) 0 0
\(655\) −1210.00 −1.84733
\(656\) 0 0
\(657\) −954.000 −1.45205
\(658\) 924.000 1.40426
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1098.00 −1.66112 −0.830560 − 0.556930i \(-0.811979\pi\)
−0.830560 + 0.556930i \(0.811979\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 630.000 0.947368
\(666\) −1188.00 −1.78378
\(667\) 0 0
\(668\) 1224.00 1.83234
\(669\) 0 0
\(670\) 60.0000 0.0895522
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 676.000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1162.00 −1.71134
\(680\) 240.000 0.352941
\(681\) 0 0
\(682\) 0 0
\(683\) −1226.00 −1.79502 −0.897511 − 0.440992i \(-0.854627\pi\)
−0.897511 + 0.440992i \(0.854627\pi\)
\(684\) 648.000 0.947368
\(685\) 0 0
\(686\) 686.000 1.00000
\(687\) 0 0
\(688\) 864.000 1.25581
\(689\) 0 0
\(690\) 0 0
\(691\) 1362.00 1.97106 0.985528 − 0.169511i \(-0.0542190\pi\)
0.985528 + 0.169511i \(0.0542190\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1132.00 −1.63112
\(695\) 1110.00 1.59712
\(696\) 0 0
\(697\) 0 0
\(698\) −396.000 −0.567335
\(699\) 0 0
\(700\) −700.000 −1.00000
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1188.00 1.68990
\(704\) 0 0
\(705\) 0 0
\(706\) 1332.00 1.88669
\(707\) −154.000 −0.217822
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −1380.00 −1.94366
\(711\) −1098.00 −1.54430
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −876.000 −1.22006
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −720.000 −1.00000
\(721\) 322.000 0.446602
\(722\) 74.0000 0.102493
\(723\) 0 0
\(724\) −552.000 −0.762431
\(725\) 0 0
\(726\) 0 0
\(727\) −814.000 −1.11967 −0.559835 − 0.828604i \(-0.689135\pi\)
−0.559835 + 0.828604i \(0.689135\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) −1060.00 −1.45205
\(731\) 324.000 0.443228
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −1412.00 −1.92371
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −1320.00 −1.78378
\(741\) 0 0
\(742\) 476.000 0.641509
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1212.00 1.62466
\(747\) 0 0
\(748\) 0 0
\(749\) 518.000 0.691589
\(750\) 0 0
\(751\) −1018.00 −1.35553 −0.677763 − 0.735280i \(-0.737050\pi\)
−0.677763 + 0.735280i \(0.737050\pi\)
\(752\) 1056.00 1.40426
\(753\) 0 0
\(754\) 0 0
\(755\) −110.000 −0.145695
\(756\) 0 0
\(757\) −1374.00 −1.81506 −0.907530 − 0.419988i \(-0.862034\pi\)
−0.907530 + 0.419988i \(0.862034\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 720.000 0.947368
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 408.000 0.534031
\(765\) −270.000 −0.352941
\(766\) 1212.00 1.58225
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −972.000 −1.25581
\(775\) 0 0
\(776\) −1328.00 −1.71134
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −1016.00 −1.28934
\(789\) 0 0
\(790\) −1220.00 −1.54430
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 396.000 0.495620
\(800\) −800.000 −1.00000
\(801\) 0 0
\(802\) 636.000 0.793017
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −176.000 −0.217822
\(809\) 498.000 0.615575 0.307787 − 0.951455i \(-0.400411\pi\)
0.307787 + 0.951455i \(0.400411\pi\)
\(810\) 810.000 1.00000
\(811\) 1122.00 1.38348 0.691739 − 0.722148i \(-0.256845\pi\)
0.691739 + 0.722148i \(0.256845\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 930.000 1.14110
\(816\) 0 0
\(817\) 972.000 1.18972
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 368.000 0.446602
\(825\) 0 0
\(826\) −868.000 −1.05085
\(827\) −394.000 −0.476421 −0.238210 − 0.971214i \(-0.576561\pi\)
−0.238210 + 0.971214i \(0.576561\pi\)
\(828\) 0 0
\(829\) −762.000 −0.919180 −0.459590 − 0.888131i \(-0.652004\pi\)
−0.459590 + 0.888131i \(0.652004\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 294.000 0.352941
\(834\) 0 0
\(835\) −1530.00 −1.83234
\(836\) 0 0
\(837\) 0 0
\(838\) 1564.00 1.86635
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −845.000 −1.00000
\(846\) −1188.00 −1.40426
\(847\) −847.000 −1.00000
\(848\) 544.000 0.641509
\(849\) 0 0
\(850\) −300.000 −0.352941
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 1428.00 1.67213
\(855\) −810.000 −0.947368
\(856\) 592.000 0.691589
\(857\) −1674.00 −1.95333 −0.976663 − 0.214779i \(-0.931097\pi\)
−0.976663 + 0.214779i \(0.931097\pi\)
\(858\) 0 0
\(859\) −1662.00 −1.93481 −0.967404 − 0.253238i \(-0.918504\pi\)
−0.967404 + 0.253238i \(0.918504\pi\)
\(860\) −1080.00 −1.25581
\(861\) 0 0
\(862\) −1164.00 −1.35035
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1012.00 1.16859
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1494.00 1.71134
\(874\) 0 0
\(875\) 875.000 1.00000
\(876\) 0 0
\(877\) 1746.00 1.99088 0.995439 − 0.0954002i \(-0.0304131\pi\)
0.995439 + 0.0954002i \(0.0304131\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −882.000 −1.00000
\(883\) 1734.00 1.96376 0.981880 − 0.189504i \(-0.0606880\pi\)
0.981880 + 0.189504i \(0.0606880\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −748.000 −0.844244
\(887\) −1614.00 −1.81962 −0.909808 − 0.415029i \(-0.863772\pi\)
−0.909808 + 0.415029i \(0.863772\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 776.000 0.869955
\(893\) 1188.00 1.33035
\(894\) 0 0
\(895\) 0 0
\(896\) 896.000 1.00000
\(897\) 0 0
\(898\) 444.000 0.494432
\(899\) 0 0
\(900\) 900.000 1.00000
\(901\) 204.000 0.226415
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 690.000 0.762431
\(906\) 0 0
\(907\) 1686.00 1.85888 0.929438 − 0.368979i \(-0.120293\pi\)
0.929438 + 0.368979i \(0.120293\pi\)
\(908\) 0 0
\(909\) 198.000 0.217822
\(910\) 0 0
\(911\) 1542.00 1.69265 0.846323 − 0.532670i \(-0.178811\pi\)
0.846323 + 0.532670i \(0.178811\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1752.00 1.91266
\(917\) −1694.00 −1.84733
\(918\) 0 0
\(919\) −682.000 −0.742111 −0.371055 − 0.928611i \(-0.621004\pi\)
−0.371055 + 0.928611i \(0.621004\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1396.00 1.51410
\(923\) 0 0
\(924\) 0 0
\(925\) 1650.00 1.78378
\(926\) 0 0
\(927\) −414.000 −0.446602
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 882.000 0.947368
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1514.00 −1.61580 −0.807898 − 0.589323i \(-0.799395\pi\)
−0.807898 + 0.589323i \(0.799395\pi\)
\(938\) 84.0000 0.0895522
\(939\) 0 0
\(940\) −1320.00 −1.40426
\(941\) 1702.00 1.80871 0.904357 − 0.426777i \(-0.140351\pi\)
0.904357 + 0.426777i \(0.140351\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −992.000 −1.05085
\(945\) 0 0
\(946\) 0 0
\(947\) 1606.00 1.69588 0.847941 − 0.530091i \(-0.177842\pi\)
0.847941 + 0.530091i \(0.177842\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −900.000 −0.947368
\(951\) 0 0
\(952\) 336.000 0.352941
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) −612.000 −0.641509
\(955\) −510.000 −0.534031
\(956\) 792.000 0.828452
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) −666.000 −0.691589
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −968.000 −1.00000
\(969\) 0 0
\(970\) 1660.00 1.71134
\(971\) −1438.00 −1.48095 −0.740474 − 0.672085i \(-0.765399\pi\)
−0.740474 + 0.672085i \(0.765399\pi\)
\(972\) 0 0
\(973\) 1554.00 1.59712
\(974\) 0 0
\(975\) 0 0
\(976\) 1632.00 1.67213
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −980.000 −1.00000
\(981\) 0 0
\(982\) 0 0
\(983\) 594.000 0.604273 0.302136 − 0.953265i \(-0.402300\pi\)
0.302136 + 0.953265i \(0.402300\pi\)
\(984\) 0 0
\(985\) 1270.00 1.28934
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −538.000 −0.542886 −0.271443 − 0.962455i \(-0.587501\pi\)
−0.271443 + 0.962455i \(0.587501\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −1932.00 −1.94366
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 280.3.c.a.69.1 ✓ 1
4.3 odd 2 1120.3.c.b.209.1 1
5.4 even 2 280.3.c.c.69.1 yes 1
7.6 odd 2 280.3.c.b.69.1 yes 1
8.3 odd 2 1120.3.c.d.209.1 1
8.5 even 2 280.3.c.d.69.1 yes 1
20.19 odd 2 1120.3.c.a.209.1 1
28.27 even 2 1120.3.c.c.209.1 1
35.34 odd 2 280.3.c.d.69.1 yes 1
40.19 odd 2 1120.3.c.c.209.1 1
40.29 even 2 280.3.c.b.69.1 yes 1
56.13 odd 2 280.3.c.c.69.1 yes 1
56.27 even 2 1120.3.c.a.209.1 1
140.139 even 2 1120.3.c.d.209.1 1
280.69 odd 2 CM 280.3.c.a.69.1 ✓ 1
280.139 even 2 1120.3.c.b.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.3.c.a.69.1 ✓ 1 1.1 even 1 trivial
280.3.c.a.69.1 ✓ 1 280.69 odd 2 CM
280.3.c.b.69.1 yes 1 7.6 odd 2
280.3.c.b.69.1 yes 1 40.29 even 2
280.3.c.c.69.1 yes 1 5.4 even 2
280.3.c.c.69.1 yes 1 56.13 odd 2
280.3.c.d.69.1 yes 1 8.5 even 2
280.3.c.d.69.1 yes 1 35.34 odd 2
1120.3.c.a.209.1 1 20.19 odd 2
1120.3.c.a.209.1 1 56.27 even 2
1120.3.c.b.209.1 1 4.3 odd 2
1120.3.c.b.209.1 1 280.139 even 2
1120.3.c.c.209.1 1 28.27 even 2
1120.3.c.c.209.1 1 40.19 odd 2
1120.3.c.d.209.1 1 8.3 odd 2
1120.3.c.d.209.1 1 140.139 even 2